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Given $f_n(x) = F(x) (\cos(\pi x))^n$, where $n \in \mathbb N , F: \mathbb R \to \mathbb R$ integrable.

Simple question: does the Lebesgue integral converge in $\mathbb R$? (and how to show?)

StMan
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Does $\cos^n x$ converge? (Yes). Find its limit and apply the dominated convergence theorem.

Matthew C
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  • Yes, $\cos^n (x) $ converges to $-1$. But how can I apply the dominated conv. theorem? I mean i have to find a function g s.t. $|f_n(x)| \leq g(x)$, right? – StMan Nov 13 '18 at 19:32
  • It doesn't actually. Cosine takes values between -1 and 1. Most places (almost everywhere) it is less than 1 in absolute value. Now takes powers... – Matthew C Nov 13 '18 at 19:34
  • Ok, also with powers, the maximum value is always less or equal than 1. But how can I apply the theorem, in that case? – StMan Nov 13 '18 at 19:59
  • If $|a|<1$ then $a^n \rightarrow 0$. Can you see what the (a.e.) pointwise limit of $f_n(x)$ is? – Matthew C Nov 13 '18 at 20:18
  • Ok, so if I'm not wrong, then the pointwise limit is 0, right? – StMan Nov 13 '18 at 20:30